Pythagorean Theorem Calculator

Find the missing side of a right triangle

This tool is restricted to the three side lengths only — no angles, altitude, area, or perimeter — so it's a focused, no-distraction Pythagorean theorem solver. It's pre-loaded with the 5-12-13 triple.

The Pythagorean theorem, explained

The Pythagorean theorem states that in any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse: a² + b² = c². It's named for the ancient Greek mathematician Pythagoras, though the relationship was known and used by Babylonian and Indian mathematicians centuries earlier — it is one of the oldest and most-proven theorems in all of mathematics, with hundreds of distinct known proofs.

The theorem is useful precisely because it lets you find a missing side without measuring it directly. Rearranged for each side, it becomes: c = √(a² + b²) when both legs are known; a = √(c² − b²) when the hypotenuse and one leg are known; and symmetrically b = √(c² − a²).

Worked example: the 5-12-13 triangle

Suppose you measure the two legs of a right triangle as 5 mm and 12 mm and need the hypotenuse. Squaring each leg gives 25 and 144; adding them gives 169; and taking the square root gives exactly 13 mm — c = √(5² + 12²) = √169 = 13. Because 5, 12, and 13 are all whole numbers satisfying a² + b² = c², this is itself a Pythagorean triple, the second most common one taught after 3-4-5, and this calculator's classification badge flags it automatically.

From there, every other measurement follows the same formulas used throughout this site: area = (1/2)·5·12 = 30 mm², perimeter = 5 + 12 + 13 = 30 mm (a pleasant coincidence for this particular triple, since area and perimeter share the same numeric value here, just different units), altitude to the hypotenuse = (5·12)/13 ≈ 4.615 mm, inradius = (5 + 12 − 13)/2 = 2 mm, and circumradius = 13/2 = 6.5 mm.

Is it a right triangle? The converse of the Pythagorean theorem

The Pythagorean theorem also works backwards. If the squares of the two shorter sides add up to the square of the longest side, the triangle must be a right triangle. This is the converse of the Pythagorean theorem, and it's how you check whether three lengths form a right angle:

1. Call the longest side c and the other two a and b.
2. Work out a² + b² and c².
3. If they're equal, it's a right triangle. If a² + b² > c², it's acute; if smaller, it's obtuse.

For 5, 12, 13: 5² + 12² = 25 + 144 = 169 = 13², so it is a right triangle. For 4, 5, 6: 4² + 5² = 41 but 6² = 36, so it isn't (it's acute). A quick shortcut with the calculator above: enter the two shorter sides as legs, and if the hypotenuse it returns matches your third side, the triangle is right-angled. Builders use the same idea as the 3-4-5 rule to check that corners are square.

When you'd use this calculator

Use this focused version whenever the only information you have is two of the three side lengths — for example, checking whether a rectangular frame, wall corner, or garden bed is truly square by measuring the two sides and the diagonal, or finding a ladder's required length given its base distance from a wall and the height it needs to reach. If you have an angle, an altitude, an area, or a perimeter instead, use the full right triangle calculator, which accepts any two of all eight possible known values.

Pythagorean theorem FAQ

See the full right triangle FAQ for more general questions.